Central configurations provide the only closed-form analytical solutions of the n-body problem. All possible central configurations of three bodies have been extensively studied along with the stability of the associated periodic orbits. Stable cases have been found for the Lagrangian triangle configuration, which we see occurring with the Trojan asteroids. However, the knowledge about four-body central configurations remains limited. An explicit parameterization of a family of kite shaped four-body central configurations has recently been published. The present research investigates the stability of periodic solutions provided by these central configurations. An analytical treatment of linear stability is carried out and the eigenvalues for circular periodic orbits are calculated. This is complemented with a numerical estimation of Floquet multipliers to determine the linear stability of eccentric periodic orbits. While most of the kite configurations are found to be unstable, regions of linearly stable cases are discovered for both circular and eccentric orbits. Further, numerical simulations of the non-linear system are performed as an independent approach to validate the linear stability results. Perfect agreement with the linear analysis is found, suggesting that stable kites may be observed in the universe.